$\dfrac{ -2r + 7s }{ 3 } = \dfrac{ 4r + 4t }{ 6 }$ Solve for $r$.
Answer: Multiply both sides by the left denominator. $\dfrac{ -2r + 7s }{ {3} } = \dfrac{ 4r + 4t }{ 6 }$ ${3} \cdot \dfrac{ -2r + 7s }{ {3} } = {3} \cdot \dfrac{ 4r + 4t }{ 6 }$ $-2r + 7s = {3} \cdot \dfrac { 4r + 4t }{ 6 }$ Multiply both sides by the right denominator. $-2r + 7s = 3 \cdot \dfrac{ 4r + 4t }{ {6} }$ ${6} \cdot \left( -2r + 7s \right) = {6} \cdot 3 \cdot \dfrac{ 4r + 4t }{ {6} }$ ${6} \cdot \left( -2r + 7s \right) = 3 \cdot \left( 4r + 4t \right)$ Distribute both sides ${6} \cdot \left( -2r + 7s \right) = {3} \cdot \left( 4r + 4t \right)$ $-{12}r + {42}s = {12}r + {12}t$ Combine $r$ terms on the left. $-{12r} + 42s = {12r} + 12t$ $-{24r} + 42s = 12t$ Move the $s$ term to the right. $-24r + {42s} = 12t$ $-24r = 12t - {42s}$ Isolate $r$ by dividing both sides by its coefficient. $-{24}r = 12t - 42s$ $r = \dfrac{ 12t - 42s }{ -{24} }$ All of these terms are divisible by $6$ Divide by the common factor and swap signs so the denominator isn't negative. $r = \dfrac{ -{2}t + {7}s }{ {4} }$